Simplify: `(0.00032)^((3)/(5)) xx ((8)/(125))^((-4)/(3))`

Answer

To simplify the expression \((0.00032)^{\frac{3}{5}} \times \left(\frac{8}{125}\right)^{-\frac{4}{3}}\), we will follow these steps:

### Step 1: Convert \(0.00032\) to a Fraction
We first convert \(0.00032\) into a fraction. The number \(0.00032\) can be expressed as:
\[ \frac{32}{100000} = \frac{32}{10^5} \]

### Step 2: Rewrite \(32\) as a Power of \(2\)
Next, we recognize that \(32\) can be expressed as \(2^5\):
\[ 0.00032 = \frac{2^5}{10^5} \]

### Step 3: Rewrite \(10\) in Terms of \(2\) and \(5\)
We can express \(10\) as \(2 \times 5\), so:
\[ 10^5 = (2 \times 5)^5 = 2^5 \times 5^5 \]
Thus, we have:
\[ 0.00032 = \frac{2^5}{2^5 \times 5^5} = \frac{1}{5^5} \]

### Step 4: Substitute Back into the Expression
Now we substitute this back into our original expression:
\[ (0.00032)^{\frac{3}{5}} = \left(\frac{1}{5^5}\right)^{\frac{3}{5}} = \frac{1^{\frac{3}{5}}}{(5^5)^{\frac{3}{5}}} = \frac{1}{5^{3}} = \frac{1}{125} \]

### Step 5: Simplify \(\left(\frac{8}{125}\right)^{-\frac{4}{3}}\)
Next, we simplify \(\left(\frac{8}{125}\right)^{-\frac{4}{3}}\):
\[ \frac{8}{125} = \frac{2^3}{5^3} \]
Thus,
\[ \left(\frac{8}{125}\right)^{-\frac{4}{3}} = \left(\frac{2^3}{5^3}\right)^{-\frac{4}{3}} = \frac{(5^3)^{\frac{4}{3}}}{(2^3)^{\frac{4}{3}}} = \frac{5^4}{2^4} \]

### Step 6: Combine the Results
Now we combine the results from Steps 4 and 5:
\[ \frac{1}{125} \times \frac{5^4}{2^4} = \frac{5^4}{125 \times 2^4} = \frac{5^4}{5^3 \times 2^4} = \frac{5^{4-3}}{2^4} = \frac{5^1}{2^4} = \frac{5}{16} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{5}{16} \]

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